Athletic had already spurned a series of first-half chances, and there was more wasteful finishing for the visitors after half time as the home side managed to hold onto their narrow lead.
Madrid's victory in the rearranged matchday nine clash left the league leaders on 36 points, with some breathing space ahead of second-placed Atletico Madrid and third-placed Real Sociedad, who they travel to on Saturday.
After the match, manager Carlo Ancelotti said his side had expected a stiff challenge from Bilbao.
"Football today is like that, when you play against a well-organised team, when you don't go further ahead, you always have to suffer," said Ancelotti. "It's a quality of the team, to know how to suffer, to know how to fight, have a strong mentality, never stop fighting, sometimes that's a quality too.
"We played against a team that had had two more days rest, that doesn't have European competition, so they're fresher and they enjoyed the second half more. If you make changes for more energy, the game is played like that. I'm very happy and very satisfied."
Madrid started well, looking to move the ball at pace to stretch a well-organised Athletic defence which had conceded the fewest goals in LaLiga this season.
The home defence looked vulnerable too, though, and Athletic's Inaki Williams saw a deflected shot fly inches wide before a mix-up between David Alaba and Thibaut Courtois left Williams in on goal, only for his touch to let him down.
Ancelotti had high praise for his keeper after the match: "Courtois is doing extraordinary things, that doesn't worry me. We have to improve the defensive aspect, we've suffered in two games from set pieces, in the first 15 games we hadn't conceded from a set piece. We have [Courtois], we enjoy him.
"They had chances, it's true, they played well, above all in the second half, but for 30 minutes only Real Madrid played. I think you have to have faith in the result."
A fluid passing move ended with Asensio's shot being pushed away by Simon, only for the rebound to land at the feet of Modric, whose miskick fell to Benzema for a straightforward finish.
Madrid's victory was still far from secure. Athletic substitute Oihan Sancet went close after a twisting run into the Madrid box, but was denied by Courtois, while substitute Fede Valverde made a crucial block in the closing minutes.